STA3045F: Conditional Expectation Summary

ConditionalExpectation Summary




Intuition

E X Y a random variable

thatrepresents the provisional average



E X Y
y the value of the RV E x x
block particianof r
when Y Ay
y
E X ly is constant on each block
Ay
E x ly is 6 Y measurable


E X Y is the best 6 Y measurable RV
that estimates X

Definition of conditional expectation


Let Cr 4 IP be a probability space
X E MF be a random variable

If G F sub sigma algebra of
E 4
is a
then RV Z
any
is a verson of E xi
g if

1 measurable

, g
2 Z satisfies the partialaverage property

Sa Z dip Sa X dip for A
every Eg

These versions are unique IP almostsurely
Define as E X
g
2 Conditional expectation if X
Y
E ME

If Cr 4 IP be a probability space
B X Y E MF be a random variable

then E x 14 E XI 6 Y